A) Let $f:F\rightarrow S$ be a flat proper morphism of schemes with geometrically normal fibers. Then supposedly the number of $\textbf{connected}$ components of the geometric fibers is constant. Why is this? Without some kind of vanishing of cohomology or information on the base, I don't see why this is true.

B) Furthermore, supposedly if $F$ is now a flat proper morphism with reduced, connected, nodal curves as geometric fibers, then there is a Zariski open subset of $S$ on which the fibers all have the same number of $\textbf{irreducible}$ components. Why is this?

Finally, how far can these results be generalized? For example, is B) true for any flat proper morphism?

geometricirreducible components, which remains the same in a Zariski open set, though (I think it is false otherwise, even in the situation you consider). $\endgroup$ – Damian Rössler Nov 29 '12 at 16:45